Entanglement entropy of fermions in any dimension and the Widom conjecture
Abstract
Description
We show that entanglement entropy of free fermions scales faster then area law, as opposed to the scaling $L^{d-1}$ for the harmonic lattice, for example. We also suggest and provide evidence in support of an explicit formula for the entanglement entropy of free fermions in any dimension $d$, $S\sim c(\partialΓ,\partialΩ)\cdot L^{d-1}\log L$ as the size of a subsystem $L\to\infty$, where $\partialΓ$ is the Fermi surface and $\partialΩ$ is the boundary of the region in real space. The expression for the constant $c(\partialΓ,\partialΩ)$ is based on a conjecture due to H. Widom. We prove that a similar expression holds for the particle number fluctuations and use it to prove a two sided estimates on the entropy $S$.
Final version
Final version